Bm Fi6051 Wk8 Lecture

8/14/2019 Bm Fi6051 Wk8 Lecture

1/39

Derivative InstrumentsFI6051

Finbarr MurphyDept. Accounting & FinanceUniversity of LimerickAutumn 2009

Week 8 Volatility & Exotics


2/39

We will discuss lognormality, what exactly doesthis mean and what are the implications.

We will discuss the causes of volatility and the

significance of volatility in option pricing

We will examine implied volatility and how themarket forecasts volatility

We will look at implied volatility shapes, surfacesand implied volatility smiles

Lecture Summary


3/39

This lecture also looks at non-vanilla (exotic)options. Each of the more recognised options areexamined in turn. We look at their uses anddiscuss how they might be valued.

Lecture Summary


4/39

Underlying the Black-Scholes model is theassumption that the price of stock follows alognormal random walk, aka, GeometricBrownian Motion (GBM) with drift

This approach has been found to work very wellin industrial practice

We will look at why

Random Walk


5/39

Changes in one time interval are independentover preceding time intervals

The size and direction of the change is (in some

way) random

Assumes market efficiency, I.e. the current stockprice reflects all available information and future

changes reflect future information

With Drift, refers to the fact that stock pricestend to increase over time

Random Walk


6/39

Lognormality refers to the fact that the naturallog of the changes in the stock price areassumed to be normally distributed

Why do we use the natural log and not justsimply the changes in stock price?

Non-negative

Recombining

Works well in practice

Lognormality


7/39

The relative rate at which the price of a securitymoves up and down. Volatility is found bycalculating the annualized standard deviation ofdaily change in price. If the price of a stock

moves up and down rapidly over short timeperiods, it has high volatility. If the price almostnever changes, it has low volatility

What causes volatility? Trading

News

Volatility

Source: InvestorWords.com


8/39

A stock is priced at 50, its volatility is 15% In one week, a 1-STD DEV move is

= 1.04

A stock is priced at 50, its volatility is 45%

In one week, a 1-STD DEV move is

= 3.12

Volatility

52

115.050

52145.050


9/39

How to measure volatility? We will examine data for the ISEQ taking daily

closing values for last three months 11th August 2005 to 11th October 2005

That is, 44 observations The Volatility of the stock , is given by

Where s = the standard deviation and is thetime interval (in years)

Volatility

s

=


10/39

All we need to calculate is s

Where ui is ln(Si/Si-1 )

And is the average of the uis

Volatility

( )2

11

1 = =n

i iuu

ns


11/39

A sample from the ISEQ daily returns

Volatility

i Date Close Si/Si-1 ln(Si/Si-1)

0 10-Aug-05 6,784.93

1 11-Aug-05 6,772.67 0.998193 -0.000785

2 12-Aug-05 6,739.24 0.995064 -0.002149

3 15-Aug-05 6,697.64 0.993827 -0.0026894 16-Aug-05 6,717.50 1.002965 0.001286

5 17-Aug-05 6,691.91 0.996191 -0.001658

6 18-Aug-05 6,668.44 0.996493 -0.001526

7 19-Aug-05 6,677.70 1.001389 0.000603

8 22-Aug-05 6,685.23 1.001128 0.000489

9 23-Aug-05 6,688.59 1.000503 0.000218

10 24-Aug-05 6,659.78 0.995693 -0.001875

11 25-Aug-05 6,639.90 0.997015 -0.001298

12 26-Aug-05 6,644.13 1.000637 0.000277

13 29-Aug-05 6,666.68 1.003394 0.001471

14 30-Aug-05 6,647.73 0.997158 -0.001236

15 31-Aug-05 6,677.28 1.004445 0.001926


12/39

From these (44) observations, I calculated thestandard deviation of the daily return as

S = 0.0052659

Or 0.52659%

Assuming 260 trading days in the year, theannualised volatility is given as

Volatility

2600052659.0

=

0849101.0=%49101.8=


13/39

Recall:

Where

Notice where volatility is embedded in the

equation

Volatility Use in Black Scholes

( ) ( )2100

dNKedNScrT=

( ) ( )

( ) ( )Td

T

TrKS

d

T

TrKSd

=

+

=

++=

1

2

0

2

2

0

1

2//ln

;2//ln


14/39

If the call option price c0 is known from marketdata, along with the other variables

We can reverse out the volatility impliedby themarket price.

Although it is not possible to calculate from theBlack Scholes equation, we can get the value byiterative techniques

Volatility Use in Black Scholes


15/39

At the time of writing, Apple Computers(symbol: AAPL) have just released quarterlyresults which were disappointing.Simultaneously, they announced the release to

market of a 60Gbyte iPod video recorder whichwas well received.

Implied Volatility


16/39

Notice the subsequent erratic stock price andhigh volumes

It should be interesting to calculate the implied

volatility of AAPL and compare against historicalaverages

Maybe we can spot a trading opportunity!

Implied Volatility


17/39

Ive selected a list of near-term, at-the-moneycall and put options from CBOE

Implied Volatility


18/39

Now, using MatLabs in-built Black-Scholesfunction:

[CALL,PUT] = BLSPRICE(SO,X,R,T,,Q) SO= Current Stock Price

X = Strike Price

R = Risk Free Rate

T = Time to Maturity

= Volatility

Q = Asset dividend rate

We can calculate the theoretical value of the calloptions

Implied Volatility


19/39

[CALL,PUT] = BLSPRICE(49.25,45,0.0375,(37/365),0.25,0) Call Option = 4.6385 [CALL,PUT] = BLSPRICE(49.25,50,0.0375,(37/365),0.25,0)

Call Option = 1.3081

[CALL,PUT] = BLSPRICE(49.25,55,0.0375,(37/365),0.25,0) Call Option = 0.1729

Compare these results with the actual market:

5.00, 2.30 and 0.75 respectively

The volatility we used (25%) is lower than thatimplied by the market prices

Implied Volatility


20/39

Now, using another of MatLabs in-builtfunctions, we can calculate the implied volatility:

ImpVol = BLSIMPV(SO,X,R,T,CallPx)

CallPx = The actual call option value

ImpVol = BLSIMPV(49.25,45,0.0375,(37/365),5.0)

ImpVol = 34.55% ImpVol = BLSIMPV(49.25,50,0.0375,(37/365),2.3)

ImpVol = 40.89% ImpVol = BLSIMPV(49.25,55,0.0375,(37/365),0.75)

ImpVol = 40.14%

Implied Volatility


21/39

In other words, the options prices tells us thatthe market believes volatility of Apple Sharesto be about 40%

IF, we believed that volatility was in fact lower,how could we exploit our belief and makemoney?

Implied Volatility


22/39

Notice that the implied volatility was about40%.

The underlying stock is the same and so theimplied volatilities should be the same, right?

Volatility Smile


23/39

Extend the previous graph to a 3D model withimplied volatility, moneyness and maturity

Volatility Surface


24/39

These are options on options A call (option) on a call (option)

A put on a put

A call on a put

A put on a call

Used to hedge for a certain period. E.g. Buy a 3-month compound call option on oil futures

during a particularly volatile period. Leverage on leverage

But if both options are exercised, then thepremium for the compound option will be larger

than that for a single option

Compound Options


25/39

Barrier is an umbrella name for a two differentoption types Knock-Out Calls and Puts

Down and Out

Up and Out Knock-In Calls and Puts

Down and In

Up and In

A knock-Out, causes the option to terminate ifthe underlying breaches a barrier

A knock-In, causes the option to activate if theunderlying breaches a barrier

Barrier Options


26/39

The following diagram shows a Knock-In

If the barrier was set below 90 at initiation,these would become Down and Ins

The strike of the option should be judiciously set

Barrier Options


27/39

An airline might buy a Up-And-In Call option onaviation fuel costs, cheaper than a standard call

The premium for a barrier is lower than that of a

normal option

A down-and-out call plus a down-and-in call equals? A

standard call!

AKA kick-outs, kick-ins, ins, outs, exploding options, extinguishing

options and trigger options

Barrier Options


28/39

Barrier Options are path-dependent. I.e. atexpiration, it matters howthe underlying pricearrived at the final value

Barrier options are typically priced using Monte-Carlo techniques, these will be discussed insome detail next semester in FinancialEngineering

Barrier Options


29/39

Binary Options

Binary Options


30/39

Also known as Hindsight Options There are two (call) types

Fixed Strike Payoff is dependent on the Smax -X

Floating Strike Payoff is dependent on the ST-Smin

The equivalent put options have Fixed Strike Payoff is dependent on the X-Smin

Floating Strike Payoff is dependent on the Smax -ST

Lookback Options


31/39

Consider the payback on each option using thefollowing graph

Lookback Options

SMAX

ST

SMIN

X


32/39

Shout options are very similar to lookbackoptions but the option holder shouts (decides)during the life of the option when she believesthat the market has peaked or bottomed.

These are cheaper than lookback options

But no guarantee that you can pick the market

bottom or top!

Shout Options


33/39

Asian Options have a payoff that depends on theaverage asset price during at least some part ofthe option life

Consider a pension fund manager with apredominantly equity fund. The fund is held for 5years. The manager should consider an Asianput option on the basket, this will protect the

fund from sudden drops in the fund value in thelast days of the fund

Asian Options


34/39

There are two types of Asian Options Average Rate Options

The payoff is dependent on the averageunderlying asset price for a given period

Average Strike Options The strike (and therefore payoff) is dependent

on the average underlying asset price for agiven period

Both forms can be either calls or puts

Asian options are path-dependent

Asian Options


35/39

Note how the payoff on a regular call option (ST)is less than that of the Asian payoff (ST

AVG )

described by the diagram below

Asian Options

ST

STAVG

Averaging Period


36/39

Also known as Rainbow Options The payoff is dependent on the value of a

portfolio of assets (e.g., stocks)

The volatility of the basket is dependent on the

price correlation of the constituent stocks Lower correlation implies higher variability

(volatility)

Typical baskets belong to fund managers orindex of shares (FTSE, ISEQ, etc)

Basket Options


37/39

An exotic option is a non-standard option

Weve looked at the more common exoticoptions

An option can be contrived to suit a particularclients needs

The pricing of these options is a financialengineering skill. The construction/engineeringof the options is a much prized skill used instructured products desks in investment banks

Other Exotic Options


38/39

Hull, J.C, Options, Futures & Other Derivatives,2009, 7th Ed. Chapter 13, 18, 24

Further reading


39/39

Hull, J.C, Options, Futures & Other Derivatives,2005, 6th Ed. Chapter 13

Further reading

Bm Fi6051 Wk8 Lecture

Documents

Transcript of Bm Fi6051 Wk8 Lecture